on splines to determine the phase of interference fringes, and thus
the deflection. Computations were performed on a PC with LabView.
In this paper, we propose a new approach based on the least square
- methods and its implementation that we developped on a FPGA, using
+ methods and its implementation that we developed on a FPGA, using
the pipelining technique. Simulations and real tests showed us that
this implementation is very efficient and should allow us to control
a cantilevers array in real time.
\section{Introduction}
Cantilevers are used inside atomic force microscope (AFM) which provides high
-resolution images of surfaces. Several technics have been used to measure the
-displacement of cantilevers in litterature. For example, it is possible to
+resolution images of surfaces. Several techniques have been used to measure the
+displacement of cantilevers in literature. For example, it is possible to
determine accurately the deflection with different mechanisms.
In~\cite{CantiPiezzo01}, authors used piezoresistor integrated into the
cantilever. Nevertheless this approach suffers from the complexity of the
microfabrication process needed to implement the sensor in the cantilever.
In~\cite{CantiCapacitive03}, authors have presented an cantilever mechanism
-based on capacitive sensing. This kind of technic also involves to instrument
-the cantiliver which result in a complex fabrication process.
+based on capacitive sensing. This kind of technique also involves to instrument
+the cantilever which result in a complex fabrication process.
In this paper our attention is focused on a method based on interferometry to
measure cantilevers' displacements. In this method cantilevers are illuminated
The overall process gives accurate results but all the computations
are performed on a standard computer using LabView. Consequently, the
main drawback of this implementation is that the computer is a
-bootleneck. In this paper we propose to use a method based on least
-square and to implement all the computation on a FGPA.
+bottleneck. In this paper we propose to use a method based on least
+square and to implement all the computation on a FPGA.
The remainder of the paper is organized as follows. Section~\ref{sec:measure}
describes more precisely the measurement process. Our solution based on the
%% qu'elle est.
In order to develop simple, cost effective and user-friendly cantilever arrays,
-authors of ~\cite{AFMCSEM11} have developped a system based of
+authors of ~\cite{AFMCSEM11} have developed a system based of
interferometry. In opposition to other optical based systems, using a laser beam
-deflection scheme and sentitive to the angular displacement of the cantilever,
+deflection scheme and sensitive to the angular displacement of the cantilever,
interferometry is sensitive to the optical path difference induced by the
vertical displacement of the cantilever.
The system build by these authors is based on a Linnick
-interferomter~\cite{Sinclair:05}. It is illustrated in
+interferometer~\cite{Sinclair:05}. It is illustrated in
Figure~\ref{fig:AFM}. A laser diode is first split (by the splitter)
-into a reference beam and a sample beam that reachs the cantilever
+into a reference beam and a sample beam that reaches the cantilever
array. In order to be able to move the cantilever array, it is
mounted on a translation and rotational hexapod stage with five
degrees of freedom. The optical system is also fixed to the stage.
where $x$ is the position of a pixel in its associated segment.
The global method consists in two main sequences. The first one aims
-to determin the frequency $f$ of each profile with an algorithm based
+to determine the frequency $f$ of each profile with an algorithm based
on spline interpolation (see section \ref{algo-spline}). It also
computes the coefficient used for unwrapping the phase. The second one
is the acquisition loop, while which images are taken at regular time
cantilever should take less than 25$\mu$s, thus 12.5$\mu$s by phase.\\
In fact, this timing is a very hard constraint. Let consider a very
-small programm that initializes twenty million of doubles in memory
+small program that initializes twenty million of doubles in memory
and then does 1000000 cumulated sums on 20 contiguous values
(experimental profiles have about this size). On an intel Core 2 Duo
E6650 at 2.33GHz, this program reaches an average of 155Mflops.
configuration is not obvious at all, it can be done via a framework, like
ISE~\cite{ISE}. Such a software can synthetize a design written in a hardware
description language (HDL), map it onto CLBs, place/route them for a specific
-FPGA, and finally produce a bitstream that is used to configre the FPGA. Thus,
-from the developper point of view, the main difficulty is to translate an
+FPGA, and finally produce a bitstream that is used to configure the FPGA. Thus,
+from the developer point of view, the main difficulty is to translate an
algorithm in HDL code, taking account FPGA resources and constraints like clock
signals and I/O values that drive the FPGA.
Indeed, HDL programming is very different from classic languages like
C. A program can be seen as a state-machine, manipulating signals that
evolve from state to state. By the way, HDL instructions can execute
-concurrently. Basic logic operations are used to agregate signals to
+concurrently. Basic logic operations are used to aggregate signals to
produce new states and assign it to another signal. States are mainly
-expressed as arrays of bits. Fortunaltely, libraries propose some
+expressed as arrays of bits. Fortunately, libraries propose some
higher levels representations like signed integers, and arithmetic
operations.
\subsection{The board}
-The board we use is designed by the Armadeus compagny, under the name
+The board we use is designed by the Armadeus company, under the name
SP Vision. It consists in a development board hosting a i.MX27 ARM
processor (from Freescale). The board includes all classical
connectors: USB, Ethernet, ... A Flash memory contains a Linux kernel
\in [0,M[$.
At first, only $M$ values of $I$ are known, for $x = 0, 1,
-\ldots,M-1$. A normalisation allows to scale known intensities into
-$[-1,1]$. We compute splines that fit at best these normalised
+\ldots,M-1$. A normalization allows to scale known intensities into
+$[-1,1]$. We compute splines that fit at best these normalized
intensities. Splines are used to interpolate $N = k\times M$ points
(typically $k=4$ is sufficient), within $[0,M[$. Let call $x^s$ the
coordinates of these $N$ points and $I^s$ their intensities.
ending with a convergence criterion, it is obvious that it is not
particularly adapted to our design goals.
-Fortunatly, it is quite simple to reduce the number of parameters to
+Fortunately, it is quite simple to reduce the number of parameters to
only $\theta$. Let $x^p$ be the coordinates of pixels in a segment of
size $M$. Thus, $x^p = 0, 1, \ldots, M-1$. Let $I(x^p)$ be their
intensity. Firstly, we "remove" the slope by computing:
We compared the two algorithms on the base of three criteria:
\begin{itemize}
-\item precision of results on a cosinus profile, distorted with noise,
+\item precision of results on a cosines profile, distorted with noise,
\item number of operations,
\item complexity to implement an FPGA version.
\end{itemize}
30 & 17.06 & 2.6 & 13.94 & 2.45 \\ \hline
\end{tabular}
-\caption{Error (in \%) for cosinus profiles, with noise.}
+\caption{Error (in \%) for cosines profiles, with noise.}
\label{tab:algo_prec}
\end{center}
\end{table}
These results show that the two algorithms are very close, with a
-slight advantage for LSQ. Furthemore, both behave very well against
+slight advantage for LSQ. Furthermore, both behave very well against
noise. Assuming the experimental ratio of 50 (see above), an error of
1 percent on phase correspond to an error of 0.5nm on the lever
deflection, which is very close to the best precision.
profile with $N=10$ that leads to the biggest error. It is a bit
distorted, with pikes and straight/rounded portions, and relatively
close to most of that come from experiments. Figure \ref{fig:noise60}
-shows a sample of worst profile for $N=30$. It is completly distorted,
+shows a sample of worst profile for $N=30$. It is completely distorted,
largely beyond the worst experimental ones.
\begin{figure}[ht]
several DSP must be combined, increasing the latency.
Nevertheless, the hardest constraint does not come from the FPGA characteristics
-but from the algorithms. Their VHDL implentation will be efficient only if they
+but from the algorithms. Their VHDL implementation will be efficient only if they
can be fully (or near) pipelined. By the way, the choice is quickly done: only a
small part of SPL can be. Indeed, the computation of spline coefficients
implies to solve a tridiagonal system $A.m = b$. Values in $A$ and $b$ can be
computed from incoming pixels intensity but after, the back-solve starts with
-the lastest values, which breaks the pipeline. Moreover, SPL relies on
+the latest values, which breaks the pipeline. Moreover, SPL relies on
interpolating far more points than profile size. Thus, the end of SPL works on a
larger amount of data than the beginning, which also breaks the pipeline.
double precision values by a power of two, keeping the integer
part. For example, all values stored in lut$_s$, lut$_c$, $\ldots$ are
scaled by 1024. Since LSQ also computes average, variance, ... to
-remove the slope, the result of implied euclidian divisions may be
+remove the slope, the result of implied Euclidean divisions may be
relatively wrong. To avoid that, we also scale the pixel intensities
-by a power of two. Futhermore, assuming $nb_s$ is fixed, these
+by a power of two. Furthermore, assuming $nb_s$ is fixed, these
divisions have a known denominator. Thus, they can be replaced by
their multiplication/shift counterpart. Finally, all other
multiplications or divisions by a power of two have been replaced by
left or right bit shifts. By the way, the code only contains
-additions, substractions and multiplications of signed integers, which
+additions, subtractions and multiplications of signed integers, which
is perfectly adapted to FGPAs.
As said above, hardware constraints have a great influence on the VHDL
to communicate between i.MX and others components. It is mainly used
to start to flush profiles and to retrieve the computed phases in RAM.
-Unfortunatly, the first designs could not be placed and route with ISE
+Unfortunately, the first designs could not be placed and route with ISE
on the Spartan6 with a 100MHz clock. The main problems came from
routing values from RAMs to DSPs and obtaining a result under 10ns. By
the way, we needed to decompose some parts of the pipeline, which adds
profile image very quickly. Currently we have performed simulations
and real tests on a Spartan6 FPGA.
-In future work, we want to couple our algorithm with a high speed camera
-and we plan to control the whole AFM system.
+In future work, we plan to study the quantization. Then we want to couple our
+algorithm with a high speed camera and we plan to control the whole AFM system.
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